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This is the access page to the free molarity calculators designed by Dr Michael Mullan. It is not unusual for students and others to miscalculate the volumes of solutions or the weight of compounds required to produce solutions. The molarity calculators accessed here should enable students and others to check their calculations.

The calculator below is based on a model developed by Giles and Lawrence (1973) to predict the grade value of Cheddar cheese. Instructions on how to use the calculator are given below. Note that the  pH and other values should be obtained from 24-hour old cheese.

Ice Cream Mix Calculator

The objective in calculating ice cream mixes is to turn a formula into a recipe based on the intended ingredients and the mass or volume of mix required.  The recipe is then processed to obtain ice cream for distribution and sale.  In the UK and in North America the formula is given as percentages of fat, milk solids-not-fat (MSNF), sugar, stabilisers-(stabilizers in the US) and emulsifiers. Since several ingredients may be available to supply these components e.g. MSNF available ingredients are selected on the basis of quality and cost.

In mathematics, science and engineering students frequent have to work with very small, e.g. 0.000005, and very large, e.g. 3200000000 numbers. For example, students in microbiology are often required to write the number of colony forming units (CFU)/mL or gram in scientific notation. To avoid dealing with these large and small numbers mathematicians, scientists and engineers have developed a particular way of expressing numbers; this is called scientific notation.

Because the concentration of fat and protein in milk vary with season, and other well documented factors, it is necessary to adjust the composition of milk for use in the manufacture of cheese and other dairy products. This process, called milk standardisation (standardization in the US) is designed to ensure that product quality is maintained at a consistent level throughout the year. This area will be dealt with in more detail elsewhere in this website.

Increasingly standardisation is being done automatically using on line instrumentation. In small dairies this is still done 'manually' including calculating the volumes (or weights) of skim milk and whole milk required to produce a particular quantity of milk of a defined fat concentration.

One of the simplest methods of routinely adjusting the fat concentration of milk for cheese, ice cream or whole milk powder manufacture is to use the Pearson Square or Rectangle. This method may also be called Pearson's Square or Pearson's Rectangle. This is a simplified method for solving a two variable simultaneous equation. While it is being used here with milk it is is a tool that can be used to help processors calculate the amounts of two components that need to be mixed together to give a final known concentration. For example, it can be used to calculate the amounts of fruit juice and sugar syrup to be mixed to make a fruit squash or fruit pulp and sugar to make a jam. It is also used in mixing rations for animal feeding and in the meat industry to produce meat products e.g. sausages to a particular fat content. Wines and other alcoholic beverages are also blended to give products of a specified alcohol concentration.New gif Microsoft Excel spreadsheets for undertaking these calculations can be downloaded.New gif

This tool can only be used for blending two components. When more than two components are involved, more complex mass balance equations have to be used. The first step in using this method is to draw a rectangle. At the centre of the rectangle write the concentration of fat required in the cheese milk. At the upper left hand side write the % fat concentration of the milk; the most concentrated fat source used. At the bottom left hand corner place the fat concentration of the skim-milk used. On the top right hand side write 'parts milk' and on the bottom left hand side write 'parts skim'.

The 'parts milk or skim' are obtained by subtracting the lowest value number, working diagonally, from either the desired final fat concentration in the case of milk or by subtracting the value for final fat concentration from the concentration of fat in the milk.

This process gives the proportions of milk and skim that must be mixed together to give the desired fat concentration. Knowing the weight or volume of the final mix, the actual quantities of milk and skim required can be obtained by a simple proportional calculation. Note this method can also be used to standardise protein, SNF and/or casein in milk.

Browsers can test their understanding of this basic calculation by using the calculator below. More information on Milk Standardisation is available in the Answers to cheese science and technology self assessment section.

Click here to use the calculator

How to cite this article

Mullan, W.M.A. (2006). [On-line]. Available from: . Accessed: 1 December, 2015.


Click here to use the calculator. Unlike most calculators on non-academic sites, the Dairy science calculator enables energy contribution from the the alcohol, organic acid and artificial sweetener components to be estimated.

Model probability of detecting a pathogen in food.

Despite the global use of HACCP systems and a legal requirement for the use of HACCP in many jurisdictions' food poisoning remains an endemic problem and large numbers of people continue to be hospitalised, die and as a result companies either face substantial legal costs and / or in many cases are forced to cease trading.

While the use of HACCP systems significantly reduces the need for microbiological end point testing of foods, sampling schemes and microbial analysis have important roles in system validation and quality assurance.

This raises an issue concerning the adequacy of sampling schemes and microbial analysis in commercial food manufacture.


This article is available On Line to enable editing. Its draft status is scheduled to be removed by September, 2015. In the meantime comments are invited from scientists and technologists familiar with the subject area to help improve this article. I wish to acknowledge the generous comments and advice received from workers in this area. As a result of the feedback I will include more information about the principles behind modelling bacterial growth and the limitations of models.

Considerable effort has been devoted to modelling the growth of pathogenic and spoilage bacteria in food. This is referred to as predictive microbiology. Mathematical equations are used to describe the effect of environment, for example temperature, or in more complex models temperature, pH, available water (Aw) and other factors that affect microbial behaviour.

The advantages of modelling have been described in the article on "Modelling in Food Technology" including reducing the costs and time required in determining the safe shelf life of new products or in undertaking pathogen challenge testing. However, caution and scientific expertise are required e.g. the Food Safety Authority of Ireland (2012) has cautioned the food industry on the use of predictive models.

The purpose of this article and supporting material is to illustrate how published research in predictive microbiology can be used in practice. This is an area in which Food Science and Food Technology undergraduates sometimes find difficulty. Part of the difficulty may exist because the steps in the calculations involved are not usually presented.

This article describes a model for the growth of salmonella on cut tomatoes and a calculator where you can enter the initial numbers of salmonella, the incubation temperature and the incubation time to obtain a prediction of final numbers.

Modelling microbial growth

Because of the critical importance of temperature early work attempted to modify the Arrhenius Law to describe microbial growth but this was either unsuccessful or the relationships derived were generally too complex for routine use. In a classic paper, Australian workers (Ratkowsky et al., 1982) proposed a relatively simple, two–factor empirical equation (equation 1) to describe the influence of temperature on microbial growth up to the maximum growth temperature of an organism, Tmax.  This is often called the square root model.

Cakes are classified as intermediate moisture foods and may be subject to spoilage by moulds. Water activity (aw), the water that is available, unbound or free, for chemical reactions and microbial growth is a major factor that can be utilised to limit or prevent microbial growth. Cakes generally have aw values ranging from 0.65-0.9.

Mould spoilage on cakes tend to occur on the surface and work has been done to model the water vapour pressure above cakes with temperature to derive models for the mould free shelf life (MFSL) of these products. The water vapour pressure above a food is determined by several factors including temperature, the water content of the food, the solutes present and the water activity in the food.

All foods have their own equilibrium relative humidity (ERH). This is the humidity at a given temperature at which the food will neither lose nor absorb moisture to or from the atmosphere.

If the food is held below its ERH it will lose moisture and become drier; above this value, it will absorb moisture from the atmosphere.  The gain or loss of water can have a major effect on a food and can influence shelf life significantly. The EHR is determined by exposing the food to carefully controlled atmospheres containing defined water vapour pressures generated using for example standard solutions of salts.

EHR and water activity (aw) are closely related. Water activity represents the ratio of the water vapour pressure of a food to the water vapour pressure of pure water under the same conditions.  Water activity is expressed as a fraction. If this is multiplied by 100 then ERH is obtained.  Most bacteria cannot grow below an aw of 0.86 (86 % ERH).

Cauvain and Seiler (1992) found that the logarithm of the MFSL had a linear relationship with EHR over the range 74-90% at 21° and 27°C.  The equations derived (equations 1 and 2):

Equation 1. Log10 (MFSL, days at 27°C) =6.42 - (0.065 x ERH%)

Equation 2. Log10 (MFSL, days at 21°C) =7.91 - (0.081 x ERH%)

can be used to determine the shelf life of new cake products rapidly and inexpensively. These equations are available in expensive commercial software for determining the MFSL of cakes.

Determine the mould free shelf life of cakes.

Literature Cited

Cauvain, S.P. and Seiler, D.A.L. (1992). Equilibrium relative humidity and the shelf life of cakes. FMBRA Report No. 150, CCFRA, Chipping Campden, UK.

How to cite this article

Mullan, W.M.A. (2015). [On-line]. Available from: . Accessed: 1 December, 2015.  

Modelling the spoilage of pasteurized milk

Spoilage of pasteurized milk is almost always due to the growth of microorganisms. These are generally introduced after heat treatment and are referred to as post process contaminants (PPCs).

The shelf life of pasteurized milk is largely dependent on the number of PPCs and storage temperature (e.g. Muir, 1996). Muir (1996) has described a simple equation (equation 1) relating the number of number of PPCs and storage temperature to shelf life of pasteurised milk.

Equation 1. Shelf life (h)={0.00621*(T+273-(269.55-0.74))*(CFC15)-0.11 x ( CFC15) x 2} -2.

Where T = storage temperature in °K; CFC15, =log10 count after pre-incubation of pasteurized milk at 15°C for 24 hours enumeration on milk agar containing a selective supplement for pseudomonads called cetrimide-fucidincephaloridine (CFC).

Muir (1996) has explained that the equation can predict shelf life at storage temperatures between 6°C and 14°C to within 1 day for between 60 and 90% of samples. The accuracy of the equation has been reported to increase as the storage temperature of the pasteurized milk increases.

Go to Shelf Life of Pasteurized Milk Calculator .

Literature cited
Muir, D.D. (1996) The shelf-life of dairy products: 2. Raw milk and fresh products. Journal of the Society of Dairy Technology. 49, 44-48.

How to cite this article

Mullan, W.M.A. (2015). [On-line]. Available from: . Accessed: 1 December, 2015.  

Commercial cheese correctly manufactured with pasteurised milk and lactic starter cultures has a well deserved reputation as a nutritious and safe product. However, under certain circumstances cheese may support the growth of food poisoning bacteria or serve as a ‘vehicle’ for their transmission.

Four pathogens are of particular significance, Listeria monocytogenes, Salmonella species, enteropathogenic Escherichia coli and Staphylococcus aureusListeria monocytogenes, the causal agent of listeriosis, is arguably the most significant of this group.

L. monocytogenes is particularly significant since it can grow / survive for long periods in cheese and cause serious illness leading to death; the death rate arising from listeriosis can exceed 30%. It can also induce abortion in humans and its ability to cross the placenta, and access the brain makes it a particularly dangerous pathogen.

 This article provides an introduction to the binary and ordinal logistic regression models developed by Bolton and Frank (1999) for predicting the probability of L. monocytogenes growing in cheese after 42 days storage at 10°C.

Characteristics of Listeria monocytogenes

L. monocytogenes is a Gram-positive, non-sporing bacterium that can grow in high salt environments (up to 10 % sodium chloride), and over a wide pH (5.0-9.6) and temperature range (< 3° – 45°C); it can grow aerobically and microaerophilically ( Bajard et al., 1996; Pearson and Marth, 1990).

While the organism is relatively sensitive to heat there has been considerable debate regarding its sensitivity to pasteurisation.  While   D 71-72°C  values have generally been reported as < 4s, a D72°C value of 4.6 ± 0.5 s has been reported (Bunning et al., 1992) for heat-shocked cells.  Assuming a D72°C value of 5s and raw milk containing 1000 (a high value) CFU / ml, pasteurisation at 72°C for 15s would be predicted to result in around 1 CFU / ml surviving (see emphasising the critical importance of ensuring  only low concentrations of this pathogen in raw milk.

Commercially, cold-filled acidic pickles, sauces (e.g. salad cream, mayonnaise) and food dressings are preserved, and their microbiological safety assured, by the use of acetic acid, salt (NaCl) and sugar. This article provides an overview of a preservation model and access to the model to enable the effect of sauce components and pH to be investigated.

The Comite´ des Industries des Mayonnaises et Sauces Condimentaires de la Communaute´ Économique Européenne (CIMSCEE) has provided guidance on a safety value, Σs, for a microbiologically safe product preserved using acetic acid (Anonymous, 1993). A safe product has been defined as one which is so formulated that when an inoculum of viable cells of  Escherichia coli is  added to the product this is reduced by 3 log cycles in less than 72 h. Products exhibiting this level of antibacterial activity have a CIMSCEE safety value (Σs) of greater than 63. Σs is calculated using equation 1: Σs =15.75 (1 - ɑ) (total acetic acid %) + 3.08 (salt %) + (hexose %) + 0.5 (disaccharide %) + 40 (4-pH).

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